Variance renormalisation in regularity structures -- the case of $2d$ gPAM

TL;DR

This paper studies variance renormalisation for the 2D generalized parabolic Anderson model driven by rough noise, introducing models that lift zero noises to nontrivial models and proving convergence via graphical computations.

Contribution

It develops a novel variance renormalisation approach for singular SPDEs where traditional methods fail, specifically for the 2D gPAM driven by very rough noise.

Findings

Established convergence of the BPHZ model over vanishing noise.

Introduced models lifting zero noises to nontrivial models.

Applied graphical computations to demonstrate convergence.

Abstract

We consider the variance renormalisation of a singular SPDE for which a Da Prato-Debussche trick is not applicable. The example taken is the $2$-dimensional generalised parabolic Anderson model (gPAM), driven by a much rougher than white noise, necessitating both a multiplicative and an additive renormalisation. To handle the discrepancy between the regularity structures of the approximate and the limiting equations, we consider models that lift $0$ noises to nontrivial models, in analogy with ``pure area'' from rough paths. The convergence to such a model is shown for the BPHZ model over the vanishing noise via graphical computations.